On Top-$k$ Selection From $m$-Wise Partial Rankings via Borda Counting

We analyze the performance of the Borda counting algorithm in a non-parametric model. The algorithm needs to utilize probabilistic rankings of the items within
$m$
-sized subsets to accurately determine which items are the overall top-
$k$
items in a total of
$n$
items. The Borda counting algorithm simply counts the cumulative scores for each item from these partial ranking observations. This generalizes a previous work of a similar nature by Shah
et al.
using probabilistic pairwise comparison data. The performance of the Borda counting algorithm critically depends on the associated score separation
$\Delta _{k}$
between the
$k$
-th item and the
$(k+1)$
-th item. Specifically, we show that if
$\Delta _{k}$
is greater than certain value, then the top-
$k$
items selected by the algorithm is asymptotically accurate almost surely; if
$\Delta _{k}$
is below certain value, then the result will be inaccurate with a constant probability. In the special case of
$m=2$
, i.e., pairwise comparison, the resultant bound is tighter than that given by Shah
et al.
, leading to a reduced gap between the error probability upper and lower bounds. These results are further extended to the approximate top-
$k$
selection setting. Numerical experiments demonstrate the effectiveness and accuracy of the Borda counting algorithm, compared with the spectral MLE-based algorithm, particularly when the data does not necessarily follow an assumed parametric model.